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Impulse-Invariant Design

Module by: Douglas L. Jones

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Pre-classical, adhoc-but-easy method of converting an analog prototype filter to a digital IIR filter. Does not preserve any optimality.

Impulse invariance means that digital filter impulse response exactly equals samples of the analog prototype impulse response: ∀n:hn= h a nT n h n h a n T How is this done?

The impulse response of a causal, stable analog filter is simply a sum of decaying exponentials: H a s= b 0 + b 1 s+ b 2 s2+...+ b p sp1+ a 1 s+ a 2 s2+...+ a p sp= A 1 s− s 1 + A 2 s− s 2 +...+ A p s− s p H a s b 0 b 1 s b 2 s 2 ... b p s p 1 a 1 s a 2 s 2 ... a p s p A 1 s s 1 A 2 s s 2 ... A p s s p which implies h a t= A 1 ⅇ s 1 t+ A 2 ⅇ s 2 t+...+ A p ⅇ s p tut h a t A 1 s 1 t A 2 s 2 t ... A p s p t u t For impulse invariance, we desire hn= h a nT= A 1 ⅇ s 1 nT+ A 2 ⅇ s 2 nT+...+ A p ⅇ s p nTun h n h a n T A 1 s 1 n T A 2 s 2 n T ... A p s p n T u n Since A k ⅇ s k Tnun≡ A k zz−ⅇ s k T A k s k T n u n A k z z s k T where |z|>|ⅇ s k T| z s k T , and Hz=∑k=1p A k zz−ⅇ s k T H z k 1 p A k z z s k T where |z|>maxk{|ⅇ s k T|} z k s k T .

This technique is used occasionally in digital simulations of analog filters.

Exercise 1

What is the main problem/drawback with this design technique?

Solution

Since it samples the non-bandlimited impulse response of the analog prototype filter, the frequency response aliases. This distorts the original analog frequency and destroys any optimal frequency properties in the resulting digital filter.

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